Analytical mechanics problem

In summary, the conversation discusses the movement of a cylindrical shell with a particle inside on a horizontal plane. The potential energy and kinetic energy equations are presented and the use of two generalized coordinates is suggested. The discussion also includes the Lagrangian and Hamiltonian equations and a solution is eventually found.
  • #1
Azael
257
1
As in the attached picture.
A cylindrical shell with mass M can roll without gliding on a horizontal plane

In the cylindrical shell a particle ,p, with mass m can glide without friction.

At the beginning there is no motion and the angle to the particles position is [tex] \phi=\frac{\pi}{2} [/tex]


I am suposed to find the movement of the center of the circle as a function of the angle [tex] \phi [/tex]

Im not sure how I should start.

The potential energy of the system is, if I place the plane of reference on the level of the horizontal plane.

[tex] V=mgR(1-cos\phi) [/tex]

Now this problem obviously only has one degree of freedom and that is the angle [tex] \phi [/tex]

But if I want to construct a lagrangian I must use two degres of freedom. The rotation angle of the cylinder [tex] \alpha [/tex] and the angle to the particle p [tex] \phi [/tex]. Because I don't know how [tex] \alpha [/tex] and [tex] \phi [/tex] are connected. Is this the correct thinking?

In that manner I get this equation

[tex] L=T-V=\frac{3}{4}MR^2\dot{\alpha}^2+\frac{1}{2}m[\dot{\alpha}^2 R^2 + R^2\dot{\phi}^2 + \dot{\alpha}^2 R^2 sin\phi ] - mgR(1-cos\phi ) [/tex]

Should I use this and solve the two lagrande equations

[tex] \frac {d}{dt} \frac{dL}{d\dot{\alpha}}-\frac{dL}{d\alpha}=0 [/tex]

[tex] \frac {d}{dt} \frac{dL}{d\dot{\phi}}-\frac{dL}{d\phi}=0 [/tex]

Im not sure if this will give me any answere though? It feels like I should express [tex] \alpha [/tex] in [tex] \phi [/tex] before I even do the lagrangian. Am I on the right track?
 

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  • #2
I assume that I in the same manner can get the hamiltonian since its a conservative force situation and holonomic constrains.

[tex] H=T+V=\frac{3}{4}MR^2\dot{\alpha}^2+\frac{1}{2}m[\dot{\alpha}^2 R^2 + R^2\dot{\phi}^2 + \dot{\alpha}^2 R^2 sin\phi ] + mgR(1-cos\phi ) [/tex]

But I am not quite sure what I would do with that one either. I know that

[tex] \dot{\alpha}=\frac{dH}{dP_{\alpha}}[/tex]
and
[tex] -\dot{P_{\alpha}}=\frac{dH}{d\alpha} [/tex]

But that doesn't seem to be much help either??
 
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  • #3
I don't know...I wish I could help...

It seems like you really do need two generalized coordinates. I mean, the particle is free to move; it doesn't seem like it is constrained in any way such that [tex]\phi[/tex] and [tex]\alpha[/tex] would be related.
 
  • #4
I think that there is an error in your Lagrangian: Since you are interested in the movement of the cylinder and watch it in the "labatory system", so to say, you can't say that the velocity of the particle in y-direction is [itex] R \dot \phi \cos(\phi)[/itex]. [itex] \phi [/itex] is fixed in the "cylinder system", so you have you add the y-velocity of the cylinder. If you square this, you should get a "mixed term" so that the both velocities are related.
 
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  • #5
I managed to solve it. my expression for the Y position of the particle was

[tex] Rsin\phi + R\alpha [/tex]

So velocity was

[tex] R\dot{\phi}cos\phi + R\dot{\alpha}[/tex]

I used the fact that the conjugated momentum is constant for the alpha coordinate since there is no alpha, just alpha dot, dependence :) tricky problem.
 

1. What is analytical mechanics?

Analytical mechanics is a branch of physics that applies mathematical methods and equations to study the motion of particles and rigid bodies. It seeks to understand the forces and energies involved in the movement of objects and how they affect their behavior.

2. What is the difference between analytical mechanics and classical mechanics?

Classical mechanics is a broader term that includes both analytical mechanics and Newtonian mechanics. Analytical mechanics uses more advanced mathematical techniques, such as calculus and Lagrangian equations, to analyze the motion of objects, while classical mechanics focuses on the basic laws of motion and their applications.

3. What is a typical analytical mechanics problem?

A typical analytical mechanics problem involves determining the position, velocity, and acceleration of an object or system under the influence of various forces. This could include finding the motion of a projectile, the orbits of planets, or the behavior of a pendulum.

4. What are the key principles of analytical mechanics?

The key principles of analytical mechanics include the conservation of energy, the principle of least action, and Lagrange's equations of motion. These principles allow us to describe and predict the behavior of a system by considering the forces and energies involved.

5. How is analytical mechanics used in real-world applications?

Analytical mechanics has many practical applications, including in engineering, astrophysics, and aerospace. It is used to design and analyze structures, predict the orbits of satellites and planets, and understand the behavior of complex systems. It also plays a crucial role in developing new technologies and solving real-world problems.

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